Would a potential publisher have a problem if you sub to them, and you had also participated in this critique/feedback for the same manuscript?
Can't imagine why. For example the Dear Author blog has two "First Page" open critiques on the weekend, where writers post their unpublished first page and anyone can comment and critique it. Naturally the writers are planning to go on an submit their manuscripts to publishers eventually. This is just a rather public way of getting a critique.
One page or 500 words is such a small portion of a novel and anyway will probably substantially change as a result of the crit that a publisher is hardly going to consider it already published. I'd think they'd be glad you've sought out critique and hopefully used it to improve the MS before sending it.
A publisher doesn't expect to be the one and only place you've submitted to. They may not want to be told "You're the 30th place I've submitted to." But they know they might not be the first and only.
My friend took part in a similar kind of event through Mills & Boon, first chapter submitted for public vote and comment kind of thing. She later submitted the story to other publishers who weren't M&B and it was published. So the fact Chapter 1 had been part of that contest didn't mean anything.
