Here's what got me thinking about this:
During the battle on Takodana, Finn takes a blaster rifle from a dead or incapacitated Stormtrooper and shows off his incredible shooting skills (Apparently he washed out from the Imperial Stormtrooper Marksmanship Academy.) He knows those F-11D blasters inside and out and can use one like it's an extension of his own body. So why didn't Finn keep that blaster?
I realized that it's because that blaster is a quite visible symbol of Finn's past identity as a trooper, an identity that is behind him, and he wishes to leave it that way.
However, he never got a chance to keep the cat from getting out of the bag even if he might have wanted to. Poe Dameron likely told the whole Resistance about Finn and his heroism, and besides he would have had to reveal his past anyway in order to establish his credibility when talking about Starkiller Base. So it's no secret that he's a former Stormtrooper.