I was taking my son hunting (watching only) when he was 4 years old, mostly squirrel hunting like you. He loved it. Today he is 32 years old and my best hunting buddy.
He didn't start actually shooting until 9 years old -- again squirrels. I wanted him to be able to hold and safely operate the gun. In fact, for two years I didn't take a gun as I focused entirely on his safety and education in the squirrel woods.
He began shooting a bow at 4 and going to 3-D shoots at 5. Indaina State champ at 8 years old (Bowhunter Round-Up 12 and under). He was "ready" to bow hunt deer when he was 12 but we lived in Kansas at that time and the law didn't allow a youngster (at that time) under 14 to hunt deer. We moved to Missouri and at age 13 he bow-killed the first deer he shot at on opening evening. I witnessed the whole thing from 20 yards away.
My grandson has been a different matter. He sat in a ground blind (went to sleep) at 4 years old. At 5 years old he was in my ground blind when the largest live deer I ever saw in KY walked by -- my buck tag was already filled. He needed to go home after less than an hour in the blind. He had very little interest in the hunt.
He is 8 this year and he'll be going along again as an observer.
The youngster must want to go. He/she must be kept comfortable. Try to prevent boredom from dominating the experience -- be ready to end the hunt before that happens.
Talking to the youngster about possible outcomes of the hunt (see nothing, miss a shot, kill the animal, wound, etc.)is important. I wouldn't force any part of the experience, especially field dressing on the youngster until they are ready (when they say they are, not when we think they should be).
These early experiences are important. They can have much influence on the youngster's reaction to hunting in the future. You can end up closer to your children than most non-hunting parents ever imagine. Or, your child could resent hunting if he/she doesn't like it and hunting steals you from him or her.
I know for some of us it seems implausible that the hunt could be boring to our youngsters. I sometimes get bored hunting. However, I have past hunts to think about and know that at any time inaction can be interrupted with the most exciting moments of my life. The youngster doesn't have that memory bank to keep them fired up.
I've known some folks who tried to force the issue too much and turned their kids off hunting. That's too bad because hunting is a very positive life experience for most folks.
One more tip. When the child asks to go (son or daughter) make it happen as soon as you can!