A lot would depend on something you don't tell us: When you refer to this "summer program" that is morning to 2:00 weekdays -- is that all summer long, every week? Or two weeks, or four, or six?
If it's long, like six weeks or more, I might do one skip but only if you really want to and not because this woman can't think of ways to entertain her child. If there were some special event to attend with this other boy, and it was one-time-only, for instance...but otherwise, um, no, you are not there to lend your son out for entertainment value. And you point out that these boys are going to see each other over the summer anyway; it's not like your son's program is preventing the boys from meeting at all, ever!
Another thing: Some programs do have some kind of pattern that needs to be followed -- for instance, if it's the kind of program where the kids are working on something over the whole time (an ongoing craft or art project, a play where they need to work toward a final show, swimming that is supposed to improve over the summer, whatever). Then I would definitely say that my child needs to be there every day, because if he misses a day, he's missing a step in what he's supposed to be doing or learning or rehearsing.
But even if your son's program is not like that -- I would be leery of skipping unless there really was a reason. I'd have him skip in a heartbeat if, for instance, some old friends came from out of town for one day only or something like that. But this mom has overstepped a bit -- maybe not so much that she deserves the blasting that some posters on here are giving her. She might just have realized that she blew it and missed the chance to have her kid do anything, and in desperation she asked you this pretty intrusive thing. That isn't an excuse for her but it could be an explanation.
Did she say if she had even tried to get into the same program as your son, even at the last minute?