I definitely wouldn't go out with someone who you think is a feminist or someone you think is just using you for a free night out.
Equal rights is more for when you're in a steady relationship with someone where both of you contribute equally, but on first dates, the old-fashioned rule of the man paying still holds water.
That's why when I picked a first date location and time, it would be later in the evening when it was assumed they already had dinner so it was just for a drink(s). If it's not going well, spending less than $10 on a woman's drink for about a half hour of seeing if it's going to work out should not be that big of a deal to anyone.
If it looks like there's no vibe and she offers to pay, I'll still say no I got it, if they keep insisting, I'll say fine, pay for your ONE drink and the date's over. If you can tell the date is going well and you both end up ordering multiple drinks etc, then when the bill comes, and she offers to pay part of it, it could be she just doesn't want to stick you with the bill by yourself and just offer to help even though she wants to see you again.
If she doesn't offer even after multiple drinks are bought, I would never ask her to chip in. I would always pick up the tab but like I said, if she did insist on paying for her portion, whether she wants to see me again or not, I would either say no I got it or maybe just have her throw in the tip to make her feel like she's contributing to something.