It isn't (much) more work for her to withhold your state and federal taxes at the same time as she's withholding SS and Medicare so you should look into how that is done (google: household employer withhold federal taxes) then go back to her, be able to explain to her how it is done, and request that she do it.
If she absolutely will not (which is weird and would put me off the job), you'll need to save every week in a special "for my taxes" savings account. I don't know what your tax bracket is so it's impossible to say how much it will cost you, but you can look it up on a tax estimator calculator (like
http://www.calcxml.com/calculators/federal-income-tax-calculator) and make sure to save a little more than that just in case.
Your employers are not required to "help" you with your state and federal taxes. Most people have them taken out before they are even paid - so your $600 paycheck would be more like $532.74 for someone having taxes withheld. That's life in a country with paved roads, emergency services, and public schools. If your after-tax income isn't enough to support yourself, you'll need to negotiate a better hourly wage, find a new job, or take on additional work unfortunately.