There is no reason not to do them on the same day. The key is to not overdo the load. In short no more than 5 sets of 5 reps on squats and 1 set of 5 reps on the deadlift. BTW, deadlift is probably the best exercise to improve your grip.
My background is strength training, so my answer may not exactly be what you are looking for. Essentially for strength gains squats and deadlifts are excellent exercises. My program alternates two full body workouts with a day of rest in between. Squats are done every session. The upper body swaps between bench press and overhead press. The other exercise swaps between barbell rows and deadlifts (both get the back muscles).
Incorporating squats and deadlifts into a split routine is very tricky precisely because they hit so many areas on your body.
- Squats and deadlifts get your glutes, hamstrings, and core
- Squats also get your adductors and quads
- Deadlifts also get your forearms, grip, lats (great back exercise), and to a lesser degree your quads.
You have to factor in recovery time, which really depends on how much volume you are doing, and your ability to recover. A beginner recovers pretty quickly, but that is because they can't get the same volume as a very strong intermediate lifter (someone who can squat 1.5x body weight to full parallel).
Lastly, you have to make sure you do these lifts correctly or you will get injured. If you do half squats or anything less than the full range of motion (ROM) you are setting yourself up for injury at a later point. If you don't perform the steps of the deadlift correctly you can injure your back. Yet, they are very powerful lifts. Even if it's not your goal, you won't be able to help getting stronger doing these lifts.
Your program can work as long as you have a full day of rest between each workout. Other than your forearms, neither lift works the arms appreciably.
NOTE: regarding the question of them stimulating arm growth, I don't think that they do. Your muscles respond to stress that you put on them, and neither lift stresses the arms that much.