You will need:
- one JavaScript interpreter
- one DOM Level 2 Core and HTML implementation
- 500g of non-standard but commonly-used DOM extensions
- a pinch of DOM Level 2 Style (which might mean also a CSS interpreter and layout engine)
- yoghurt pots, round-ended scissors and sticky-back plastic
Once you have assembled your components (remember to get a grown-up to help you with the sandboxing), you'll find what you have is essentially indistinguishable from a web browser.
JAVA is not part of the shell build on the server. V8/SquirrelFish is C++ code I would need to convert to PHP.
Porting a JS engine to PHP would be a huge task, and the resulting performance likely horrible. You can't even really get away with a nearly-solution on JavaScript any more, since so many pages are using hideously complex libraries like jQuery to do everything, which will require in-depth JS support.
I don't think you're going to be able to do this purely in PHP. You'll have to hook up Java/Rhino/HTMLUnit or a proper web browser like Mozilla. If your hosting environment doesn't give you the flexibility you need to compile and deploy that sort of thing, you'd have to move to a better hosting setup with a shell (preferably VPS).
If you can avoid this unpleasantness some other way, by special-casing known pages' AJAX access, do that.