I think it matters, even if it's a "free" day. If you're trying to do a mental and physical discipline, looking for loopholes doesn't seem the best way to go about it.
I have had friends whose husbands are "picky eaters" like that, and the kids pick it up from dad; I hope you won't have a problem with that. I'm such a dedicated cook that I don't think I could have married a man who was a picky, non-adventurous eater! His parents obviously tolerated it when he was a kid. :[
i totally agree with you on the parents. as i said, i lay this at the feet of my mother-in-law! and my kids eat chinese, mexican, italian etc. Things their father wouldn't touch with a ten foot pole!