In this case, context is everything. The first question to ask is: "Did 'Kansas City' have any special meaning to the narrator and his father, at that time, particularly in connection with 'clutching' something?" And the answer is: "Yes!"
In the version of the book that I'm looking at right now, in Chapter 16 "the Old Man" (the narrator's father and boss) orders him to go scout out Kansas City, which is known to have aliens in it, and see what catches his eye. Don't get caught, and report back by 11:30 PM.
Therefore, Sam (to call the narrator that, since his wife does, even though he wasn't born with the name) spends Chapter 17 and most of Chapter 18 within the city limits of Kansas City. He eventually estimates that roughly 90 percent of the adult population is already enslaved, but that this doesn't show on the surface because the titans are continuing the masquerade of making people who are possessed behave "normally" in most respects.
Sam has barely arrived in the city, coming through a toll gate in his car, when he is directed by a uniformed cop to stop and take an eye exam as part of a local traffic safety drive. Knowing what he knows, Sam instantly assumes that the true purpose of the exercise is to get each new arrival in town completely off guard, staring into a device, while someone places a new titan (or "master") on the back of his neck so it can control his central nervous system. Here's how he handles this challenge. (Note: In this passage, and the next one that I quote, I am adding boldface to a key word so it catches the eye.)
The assistant had moved in closer: the cell was ready in his hands. As
I turned he tilted it away from me. "Doctor," I said, "I wear contact
lenses. Should I take them off?"
"No, no," he snapped. "Let's not waste time."
"But, Doctor," I protested, "I want you to see how they fit. Now I've
had a little trouble with this left one--" I lifted both hands and
pulled back the upper and lower lids of my left eye. "See?"
He said angrily, "This is not a clinic. Now, if you please--" They were
both within reach; lowering my arms in a mighty bear hug I got them
both -- and grabbed with clutched fingers at the spot between each set of
shoulder blades. With each hand I struck something soft and mushy
under the coats and felt revulsion shake me at the touch.
Once I saw a cat struck by a ground car; the poor thing leapt straight
up about four feet with its back arched the wrong way and all limbs
flying. These two unlucky men did the same sort of thing; they
contorted in every muscle in a grand spasm as if every motor cell in
each body had been stimulated at once.
Which is perhaps just what happened when I clutched and crushed their
masters. I could not hold them; they jerked out of my arms and flopped
to the floor. But there was no need to hold them; after that first
boneshaking convulsion they went limp, unconscious, possibly dead.
A bit later, Sam has just bought a newspaper to see what the non-possessed members of the local populace are being told about news of the outside world, and then a police officer (whom Sam is equally sure has an alien riding on his back) approaches him. Here's what happens.
He stopped in front of me. "Let me see your license," he said
"Certainly, officer," I agreed. "It's clipped to the instrument board
of my car." I stepped past him, letting it be assumed that he would
follow me. I could feel him hesitate, then take the bait. I led him
around to the far side, between my car and his. This let me see that
he did not have a mate in his car, a most welcome variation from human
practice. More important, it placed my car between me and the
"Right there," I said, pointing inside, "it's fastened down." Again he
hesitated, then looked -- just long enough for me to use the new
technique I had developed through necessity. My left hand slapped down
on his shoulders and I clutched with all my strength.
It was the "struck cat" all over again. His body seemed to explode, so
violent was the spasm. I was in the car and gunning it almost before
he hit the pavement.
And then there's a similar moment, during this same trip, after Sam decides he needs to capture an enslaved human and bring the guy back to Washington to testify to the severity of the problem in Kansas City. Sam implicitly uses a similar technique (although the exact word "clutch" is not mentioned this time around). Note: His selected target's wife has not been taken over by an alien, so she doesn't need the same treatment.
There was a small table just out of my reach. I half rolled, half
lunged, grabbed a leg and threw it. It caught him in the face as he
was grabbing the poker. Then I was on him.
His master was dying in my fingers and he himself was convulsing under
its last, terrible command when I became aware of nerve-shattering
screams. His wife was standing in the doorway. I bounced up and let
her have one, right about her double chin. She went down in mid scream
and I returned to her husband.
The first excerpt was from Chapter 17; the other two are from Chapter 18.
It's only a bit later in the plot, in Chapter 20, when we have the scene that includes Sam using the phrase "I'll give you the Kansas City clutch." The above passages show us what it meant, to him, to clutch something in a certain way during his recent trip to Kansas City.
So I don't think "Kansas City clutch" was a common figure of speech in American pop culture when Heinlein was writing this story in the 1950s; nor do I think he meant us to assume that it was supposed to be a common figure of speech in the imaginary future which he was describing in his story. Instead, it was a phrase which Sam was inventing on the spur of the moment, knowing that the Old Man would understand the intended meaning when he considered it in the context of Sam's recent report about that scouting trip to Kansas City.
In other words, the intended meaning was approximately the following:
"I will check you over with my hands, pressing against any part of your anatomy covered by cloth. If I detect anything that feels like a big mushy lump where it really doesn't belong, I shall assume the worst and instantly crush it in my fist, right through the fabric, the same way I did on multiple occasions in Kansas City!"