Servos don't have integrated speed control, you command a position and they move there as fast as they can to reach it, but you can slow them down and still get smooth motion by changing the position signal in small increments over time rather than all at once.
You have some control over the pressure that the servo squeezes by controlling the position you send it to, and thus how much the foam is compressed. You will also want to get a servo that isn't too powerful, just in case you accidentally squeeze it all the way. I got a HS-322HD for my claw, but if you're going to be using this a whole lot I would probably go with a HS-325HB. It's basically the same servo, but its gears are made out of a more durable material that will last for more movements. At 5V it can only deliver 42 inch-ounces of torque, and if I'm doing my mental free body diagrams correctly (the radius of the output disk is about half an inch, but the torque is split between two fingers) the maximum compression would be the same as laying a 2.625 pound weight on the button. Squeezing the button that hard every time is a little excessive and could wear it out more quickly than normal use, but it wouldn't break it immediately. I just put my cell phone on a kitchen scale and apparently I mash it's buttons with about a pound of force. Wow, my poor cellphone!
As for holding the button down too long, you could control how long the servo presses the button in your microcontroller program. Each time you press the button the program would execute a close-pause-open routine, with a pause length you determine by experimenting.
If you want your button presser to be portable, you will need batteries to run it, so a battery pack in the 4-6V range would work well, like a 4 or 5 cell NiMH pack. One or two servos won't draw all that much power, so if you don't need the portability you could also use a "wall wart" style ac adapter in the 5-6V range.
You will also need some sort of microcontroller to run the show. Computationally, controlling one or two servos is pretty simple, so just about any microcontroller will do the job, so selecting one depends more on your preferences. Do you have any prior programming or electronics experience? You should probably choose a microcontroller that you can either program in a language you already know, or one that has a simple, beginner oriented system. Another question to think about is do you think this is a one-time project for you, or the beginning of a new hobby? You have a few more options if you're willing to also purchase a programmer for your microcontroller, but there are also many microcontrollers that come pre-programmed with a bootloader so you can program them directly from a computer serial (or USB) port.
I was also thinking you might want a controller that has buttons already built in, but I only know of ones that have those little tiny circuit board mounted buttons. If you're going to be using it a lot you're going to want to get some bigger, nicer buttons, so that's not really a consideration.
If this is starting to seem like a bit too much there are also standalone devices that let you record a servo motion once and play it back at the push of a button. There must be others, but the only one I know of is here. I haven't used one, so I can't vouch for it's quality, but it seems to me like it would be more than capable for your task, and as such, a little pricey. Then again, you don't have to think about programming, or soldering! It will control up to four servos, so if you wanted two claws to press either the forward or back buttons you could make one "recording" moving both servos at the same time, then wire up a toggle switch that would only power one servo or the other. You could also flip it into loop mode if you wanted to scroll through a bunch of pages automatically.
So, what sounds good to you?