Whilst I don't know the answer for sure there seems to be very little connection between the shows beyond similar generic tropes and art style. The similar art style has even been mentioned by Justin Roiland, co-creator of Rick and Morty.
Were there other shows you looked to model this after?
Harmon: No, I didn’t come to the table a huge animation buff.
Roiland: In terms of visiual look, no. People are gonna say that it’s got similarities to Regular Show, but that’s just my style, J.G. [Quintel]’s style and even like Penn Ward. You’ll notice mouths are kind of similar and teeth are similar, but I think that’s also a stylistic thing that… all of us are kind of the same age, and we’re all inspired by The Simpsons and all these other shows we’re kind of subconsciously tapping into.
Vulture, Talking to Dan Harmon and Justin Roiland About Their New Adult Swim Show, ‘Rick and Morty’
In fact when looking for connections between the shows the only thing I can really find is that Rick and Morty is an Adult Swim show which is a part of Cartoon Network which makes the Regular Show. There doesn't even appear to be any directors or writers in common between the two.
Rick and Morty first aired on December 2, 2013 and the Eggscellent Knight first appeared in the episode "Eggscellent" which first aired on February 27, 2012. Therefore, it seems highly unlikely he has a connection to Rick.
It is most likely that the Eggscellent Knight is just a reference to the Grail Knight from Indiana Jones and the Last Crusade as also mentioned on the Regular Show wikia.
The Eggscellent Knight is based on the Grail Knight from the film Indiana Jones and the Last Crusade. Aside from physical resemblance, they both guard a room of similar-looking objects (goblets in the original, hats in "Eggscellent") that visitors must choose from; those who "choose poorly" are met with death. Jonathan Kimble's fate echoes that of Walter Donovan, the primary villain of Last Crusade.
Regular Show wikia, Eggscellent Knight
Compare the rooms and knights in the following images.
u/Drew_Ferran on reddit thinks they found Rick in the Regular Show from the episode "Party Pete" and it's a closer resemblance than the Eggscellent Knight but in my opinion even less probable with "Party Pete" having first aired on February 21, 2011.