How to negotiate employer paying for flight to interview?

I would be shocked if they didn't offer to pay for the flight to interview you. The point of all the phone screens is that they feel strongly enough about you to take the financial risk to fly you out for the in-person interview. To answer the question, though: If a employer wanted to interview me in person and said that they wouldn't pay for a flight, I would tell them I'm not paying for a flight. I mean, I'd do it politely, but that's a pretty common sense line for a person searching for a job to draw. (I have never heard of a interviewee being asked to pay for a flight to a job interview.)

All the large public universities I know will pay your airfare + hotel + car rental

All academic institutions I've worked for and interviewed with have paid for the flight. It's standard procedure.

Keep good records of your costs and retain your receipts. When (and if) they invite you for a face-to-face interview, if they don't mention covering your travel expenses, you could always ask them "what sort of documentation do your reimbursement policies require?" as a way of broaching the subject. If the university's budget for hiring biomedical research technicians doesn't take into account that candidates might not reside locally, they could be prohibited from covering the cost of your flight. This would, as you noted, put you in the position of fronting the airfare without a guarantee of getting the position. However, assuming you get the job, you could (and certainly should) inquire about moving expenses.

As a datapoint, I was asked to buy my own ticket for a job interview, and subsequently was offered and took the job. In my case, they said that there was no room in the hiring budget. It was a mid-level job that could have been filled locally, but I had niche experience that made me a particularly attractive candidate. I looked at the matter as: I was the one lobbying for a relocation, while they simply looking for a good fit. For many academic positions, it's unlikely a local candidate would fall in their lap. So it makes sense for them to foot the bill. I'm not sure if that applies to research technicians. But I would factor that into the equation before automatically chalking it up as a "bad omen".

Beyond the reasons stated, if they are not willing to pay for your flight, then the job is most certainly not your ideal job that you would love to have. It signals that your employer literally has no money, which does not bode well for the future. It means their budget is so tight, they can't even cut the $500 it would cost to fly you from their expenditures. What that means is that when their budget gets cut by the academic institution in question (likely by an amount greater than $500), they'll have to do much more radical things like laying people off to balance their books.

I've worked for several labs in a large private university in a large city in the northeast and none would ever consider flying out candidates (nor moving expenses) for any of their research technician positions, especially if they were entry level and requiring only a bachelor's degree and up to 2 or 3 yrs of experience. In the labs I worked for, I don't believe that this reflected tight budgets but more that they were able to hire without needing to do so. It could certainly be different at other institutions/labs.

Also, I would be taking all the financial risk by paying with no guarantee of getting the job. Just act if its assumed that they will reimburse you for the flight. I have seen some recruits who are in a sought-after enough that this is standard and everyone knows it. Act like it. BUT...say stuff to them to ensure(insure?!?) that an agreement was reached like "Do you want a hardcopy of the tickets with price from the airline, or will an email with the details suffice?" Thats all. No need to think anymore about it. Although...maybe I'm confused, but I don't know if a big 10 school would fly you from the left coast for a tech position.

The position is a biomedical research technician at a large public university in a decent size lab that publishes regularly. You don't negotiate with state agencies. It's either in the policy or it isn't and the hiring committee will follow that. You can probably find their policies on their website. For example, ours will cover unclassified staff interviews at the rank of instructor or higher. I gather classified staff are not offered this.