Well the short answer is: Why not try it for a month and see whether or not it's something you're capable of doing? Because you won't know until you try.
One of the main concerns of practicing more than one style at a time is that you might confuse them, and it will annoy your teachers as well as perhaps slowing your progress. This is a bigger concern for styles that are very similar (to beginners anyway). For example: Taekwondo and Kyokushin Karate, or Judo and Brazilian Jiujitsu.
Eagle Claw kung-fu and kickboxing will share only some basic strike techniques. The rest is very different. The two styles are pretty far apart, and I think you won't have too much trouble compartmentalizing them and not confusing them. During sparring you might confuse them, but you can deal with that as it happens. That is to say, your teachers will deal with it.
When you're at an advanced level in one style, then even similar styles will seem very different to you, and this ceases to be an issue. With just a year of training in Eagle Claw under your belt, you're not there yet. But like I said, the two styles are different enough that I think you have a good chance at not confusing them.
Next, you indicated that you want to train both Eagle Claw kung-fu and kickboxing on the same days, spaced a half hour apart, for a total of 3 hours per day training. This can be pretty aggressive and hard on the body. The main problem to watch out for here is something known as "over-training". That's where your performance plateaus or even worsens over time, because your body isn't able to recover completely.
Assuming you have a whole day off from training (48 hours between training sessions), you have a good chance of recovering, more or less completely, from the previous day before going into the new day of training.
But you still can't go so hard that you trash your muscles each time. If you do that, it will be a week or longer for you to recover. So you have to watch this very carefully. Don't overdue it, especially not in the first month. Take it much more easily than you're used to.
Also, when you have that 30 minutes of rest time between classes, immediately drink something that has glucose and electrolytes in it. Something like Gatorade. This will help you recover from your previous class so that you won't feel as bad in your second class of the day. You're trying to replenish muscle glycogen as fast as possible. So down that sugary drink. It will help you.
Running on nights that you're not training probably won't be good for you. You're going to need those days to recover. Probably. You might limit it to one or two nights a week, and do it on a weekend or something. Or do it right after your training has completed for the days you're training.
You might also benefit from taking a week off from training once every 4 to 6 weeks. It's non-intuitive, but this will allow your body to recover and will actually cause you to make more progress over time than if you did not take those weeks off.
One other problem you might see will happen well before the effects of chronic over-training are felt: When your body is even just a little tired, your accuracy and form begins to worsen. Training "technique" (form) when your body is in this state will cause you to drill incorrect form into your muscle memory. This can hinder your progress. You always want to drill with correct form.
So if you're going to train two styles on the same day, and you have a choice of which style to train first in the day, pick the style that has a greater emphasis on refined movement and form. Because that's the class you're going in to "fresh". The second class you're going to go to in a tired state, and so it's not as good for the second class of the day.
Like I said, you won't know until you try. Give it a try. See what happens.
I've personally been in such good shape in my earlier years (ages 16 to 21) that I could handle 3 hours of class per day, 3 days a week, just fine back then. Nowadays? No way. But that's me.
Oh, and just to comment on what you're doing with regards to dropping your gym workout: My opinion is that if you're trying to keep body fat low, cardio fitness high, and overall muscle tone and flexibility good, then this martial arts program you've come up with is very good for that purpose. You don't need gym workouts for that. But one thing you can do much better at in a gym is weight training, especially with free-weights.
Building muscle and strength is very difficult to do with just martial arts training, unless those programs also have you lifting weights (most don't do that in class). You can get "toned" in martial arts training, but it's not the same as actually building muscle.
The bottom line is that this depends on your goals and what you want out of it. It also depends on your own body and how it can deal with the increased loads and stresses. Listen to your body for signs of over-training.
Hope that helps.