Assuming you could legitimately target the Eidolon with your wand (Longstrider probably isn't a valid spell, since I believe that is self only), yes, they can benefit from magic items that other players use. The rule is that the Eidolon cannot use or wear magic items (or items with an activation entry, based on comparable Familiar/Animal Companion rules) unless said items also possess the Eidolon trait.
Others characters may use magical items that can target the Eidolon, and the Eidolon can benefit from such things, merely that they cannot use them themselves. It's no different than a non-spellcaster being able to benefit from a Scroll, even though they can't use said Scroll (barring Trick Magic Item and similar feats).
So from the eidolon rules its definite that eidolons can't use or wear magic items and its possible they can't use any items at all.
So I won't go into the using all items rule again because that seems to have taken over the summoner thread but I was wandering if people agreed with my reading of the rules when it come to third parties using items to benefit the eidolon.
So their doesn't seem to rule that eidolon don't benefitting from magic items only they can't use them or wear them.
So if someone (say the summoner) was to use a wand of long strider on the eidolon it would work and they would gain the benefits from it. If someone else where to feed/activate a healing potion (or any other elixir or potion) for the eidolon that should also work as they aren't using their own actions to use these items.
Which means technically an eidolon should be able to benefit from having an alchemist in the party as long as other people spend their actions to feed/ activate/cast the items for the eidolon ?
Yes, the combo between a Summoner and an Alchemist is a very strong one. Giving Bestial Mutagen to the Eidolon improves greatly its combat abilities, reaching Fighter level of damage output if you also add Boost Eidolon.
I'm not sure what wand of longstrider is doing in there, though. Longstrider is a spell that affects the caster, not one someone else could use on an eidolon.