In reference to the patent: US20080121654
Is the whole idea of a dual-squeezable bottle have a patent or is it just this particular cap and this particular bottle?
The document you linked to is a patent application. The resulting granted patent is US7665631.
To answer your question, what the patent covers is described in its claims. In this case, there is only one independent claim:
A dual liquid condiment dispenser bottle, comprising:
a. a squeezable bottle having two separate chambers, defined by a flat wall dividing the bottle into two separate halves;
b. a dual valve housing with two rectangular slots or apertures, each rectangular slot or aperture positioned over one of said two separate chambers through which the condiments are to flow upon squeezing said liquid condiment dispenser bottle, said dual valve housing affixed to the top of said liquid condiment dispenser bottle by a press fit or mutually keyed or mated molded means;
c. two rotating valves, each one having a flexible locking tab that rides inside a mating groove inside said dual valve housing for the express purpose of retaining each said rotating valve in position to maintain alignment with a corresponding rectangular slot or aperture in each said rotating valve and said rectangular slots or apertures located in said dual valve housing;
d. a finger tab on each rotating valve to provide the external means with which to rotate each valve separately;
e. an external threaded portion molded on the exterior of the neck of the squeezable bottle; and
f. a cap having a threaded portion that corresponds to said external threaded portion on the neck of said squeezable bottle, said cap fitting over said dual valve housing.
As you can see, this patent is quite specific to the implementation described with a relatively long and complicated claim. To infringe on this patent you would need to implement each and every step in the claim.
This doesn't mean there might not be other patents that are generally relevant to dual squeeze bottles. This patent has four cited patents as prior art, although none are obviously about squeezable bottles. There are seven patents that cite this patent and may be relevant. This doesn't guarantee there aren't others. For instance, I've seen epoxy packaged in flexible pouches where there the two components are meant to be squeezed out to mix.