What exercises can I do for my posterior chain if I have shin splints? Can I squat or deadlift?
Shin splints tend only to be affected by impact exercises—running, skipping, and jumping—or exercises that load the tibialis anterior. Squats and deadlifts should neither aggravate your shin splints, nor be adversely affected by them. If you are squatting deeply, however, the acute dorsi flexion required of the ankles may cause you discomfort. If so, you can simply modify your training temporarily to perform parallel squats instead.
Lunges, step-ups, leg presses, hip thrusters, kick-backs, and most other leg exercises should also be safe. As a caveat, of course, if a movement is aggravating the shins, modify or avoid it.
Finally, if you do not already have a remedial exercise, try this: Stand on the edge of a stair facing outwards (downstairs, as it were), with your heel (calcaneal tuberosity) supported but the rest of your foot cantilevered over the edge. Your centre of mass is anterior to your tibia, so your body weight should cause your toes to drop downward over the step. Lift your toes up, and gently control their fall back their starting position. The combination of the stretch and eccentric contraction is very effective at aiding recovery.
I hope that helps.