Question regarding negotiations

Well, it is reasonable for them to confirm that you are not just making this up! Just a name won't do that of course, but most likely they want to counter with all the negatives of the other agency.

One possibility of a real negative is high-balling, not uncommon in sales. An agency may offer much more initially than they will in the contract. This "ruins" you to talk to other agencies with a real offer so of course you will be back. Of course it has a downside when they offer you less than they promise, but it is a strategy that some agencies or recruiters may use (I haven't heard of this strategy in reference to a specific agency, but of course you do read about busted compensation promises).

Before continuing, I would try to put them in a box (in sales lingo). Ask them if they will match the offer if you supply the information they want. Otherwise you have little incentive to comply but of course if this works, you win big, getting everything you want. Of course you want to emphasize that they are your first choice if their offer is closer or equal.

So I gave them the information they wanted and they verified it with the main staffer. They have responded by telling me that the other company is able to pay me more because they put less money in the hourly rate and more than make up for it with the subsidies. They said that is poor IRS practice and they have moral and legal obligations to not do that. To me the difference is small, $18/hr base rate vs $15/hr base rate. They also told me if I were to be audited I would be responsible for more taxes because it's not realistic for an RN to make $15/hr. I would believe this risk would fall on the agency and not myself, correct?

Yes, no real risk for the employee. So the real question is, was there a significant total pay difference between the two agencies in the calculator, or was the only difference was net pay?

It's about $5/hr on the calculator. I'm still new to this as this upcoming contract will only be my 3rd. However, the thing that baffles me is that the company lowballing all the offers is one of the top travel nurse companies on TravelNursingCentral while the one offering me more is more mediocre in their ratings. Do people just not shop around?

Yup, you have to shop around. Lists are only a starting place and any list is seriously flawed.