Both EPS and ABS lights are on, is there a problem with the brakes?

They will be okay once the tire is replaced. Since you are on a spare tire the diameter between the spare and your normal tires are going to differ a bit. Since the ABS and EPS systems all have sensors on each one of the four wheels, it is sensing the difference in tire diameters and disengaging the system. Your brakes are fine, you just don't have antilock so they can lock up under heavy braking.