是否可以在手机上将方形的颜色更改为绿色？

As Daan pointed you what you're really looking for to achieve something that you described are websockets. Though you could use a polling approach with jquery to do the same, it really isn't as elegant as using websockets. There are a lot of websocket libraries available for which a detailed answer is given in this stackoverflow thread

Firebase is a backend and simplifies a lot of the development for you especially to build real time applications like the one you wanted to. Have a look at Firepad. Open it up on two different browsers and you'll be able to see it's real time nature. As you edit on one browser screen, the content will appear on the other.

You can use this as a starting point, look into it's code on github and build your app. I think you could consider building apps with meteor