There is no official clarification about wild shape and synthesist on the official FAQ. But we can read both abilities to check what we can understand out of them.
From the Synthesist's Eidolon ability we have:
While fused, the synthesist loses the benefits of his armor. He counts as both his original type and as an outsider for any effect related to type, whichever is worse for the synthesist. Spells such as banishment or dismissal work normally on the eidolon, but the synthesist is unaffected. Neither the synthesist nor his eidolon can be targeted separately, as they are fused into one creature. The synthesist and eidolon cannot take separate actions. While fused with his eidolon, the synthesist can use all of his own abilities and gear, except for his armor. In all other cases, this ability functions as the summoner’s normal eidolon ability (for example, the synthesist cannot use his summon monster ability while the eidolon is present).
And from Transmutation (Polymorph):
When you cast a polymorph spell that changes you into a creature of the animal, dragon, elemental, magical beast, plant, or vermin type, all of your gear melds into your body. Items that provide constant bonuses and do not need to be activated continue to function while melded in this way (with the exception of armor and shield bonuses, which cease to function). Items that require activation cannot be used while you maintain that form. (...)
While under the effects of a polymorph spell, you lose all extraordinary and supernatural abilities that depend on your original form (such as keen senses, scent, and darkvision), as well as any natural attacks and movement types possessed by your original form. You also lose any class features that depend upon form, but those that allow you to add features (such as sorcerers that can grow claws) still function. While most of these should be obvious, the GM is the final arbiter of what abilities depend on form and are lost when a new form is assumed. Your new form might restore a number of these abilities if they are possessed by the new form.
There are three things to point out here:
- The synthesist and the eidolon are the same creature;
- Your armor melds into your body;
- You lose all extraordinary and supernatural abilities;
From my understanding, neither the caster nor the eidolon can be targeted separately (except for the spells listed on the class), meaning that if you polymorph yourself (Beast Shape), your eidolon is also targeted by the spell, as you are one creature while merged. Spells that are personal or target: you still have a target (the caster) and affect both the eidolon and the synthesist as they are one creature.
Similarly, if the synthesist is targeted by a Baleful Polymorph spell, he shouldn't be able to simply ignore the spell effects by still having access to all of his eidolon attacks and abilities.
There is at least one FAQ item in favor of treating the eidolon as your armor, and thus make it meld into your body when polymorphed.
Summoner: Does a synthesist (page 80) keep any armor bonuses or other benefits from his armor when he is fused with his eidolon?
No. The eidolon is, in effect, the synthesist's armor, and any armor the synthesist wears does not function while the eidolon is present. Fortunately, this doesn't mean the synthesist has to remove his armor when wearing the eidolon, and when the eidolon is banished/dismissed/killed, the summoner's worn armor immediately begins functioning again.
And since a synthesist can still gain all benefits of magic effects cast upon him, like Invisibility or Shield, it would make little sense to rule it differently for polymorph spells when no exception is given by the class. There is another FAQ item explaining what happens to penalties and afflictions.
an invisible summoner isn't suddenly visible if his eidolon disappears, nor does he become invisible again if the duration ran out while the eidolon was gone.
In other words, if you change shape, your eidolon's abilities should disappear from your character.
On the other hand, if you happen to summon your eidolon, a class ability that doesn't sound to be dependant on form (but GM fiat is required here) after you wild shape, your eidolon wouldn't gain any benefits from your wild shape, as he wasn't present when you targeted yourself with it. Considering the fact that you use your eidolon physical ability scores, there would be no benefit from doing this.
The question of whether or not the eidolon evolution abilities could be used should also require a GM judgement call, as many of those are clearly dependant on form, like claws and extra limbs. Not to mention that your eidolon must be at least the same size as your character for the merging to work, which kind of beats the point of wild shaping into a larger creature. You would have to pick an eidolon form that is already large or huge so you can turn into a tiger or something larger, which would be a 4-point evolution per increment in size, not available at first level (8th for large, 13th for huge). Until then, you would be stuck with a medium eidolon.