No......I dont think they would have, because men dont usually save things like that. Even with the shirts, as wonderful as that was, it was quite a bit out of the ordinary for a man to be so sentimental as to keep those shirts. In fact, I dont think this has ever been discussed, but I wonder when Jack went back to his folks place after sheepherding on Brokeback, did his mother get his clothes out to wash and ask about the blood on the shirt, and Jack maybe told her about Ennis, and it was her who hung the shirts in the closet........(maybe thats another thing that could be discussed here).
On the other hand, maybe Jack did keep some postcards from Ennis, and that is why Laureen was so snarly with Ennis when he phoned after Jack died. Maybe she had found them and that is how she worked out that Jack and Ennis were more than "just friends".
I definately dont think Ennis would have kept them, not before Jack died anyway.....he was just too scared and paranoid........but that all disappeared after Jack died when he hung the shirts and a postcard on his closet door.