Absent DM permission, probably No at level 1
Technically, to take a feat a creature must meet that feat's prerequisites when the feat is picked: "Your character must have the indicated ability score, class feature, feat, skill, base attack bonus, or other quality designated in order to select or use that feat" (Player's Handbook 87).
This means that—again, technically,—the DM must agree that when the druid 1 that possesses the alternative class feature shapeshift (Player's Handbook II 39–40) gains that first level of druid that the druid immediately assumes predator form gaining a bite attack, casts the 1st-level druid spell beast claws [trans] (Spell Compendium 25) so as to gain two claw attacks, and then—during the claws spell's 1 hour duration—picks as his feats the feat Multiattack and, for instance, as a human druid's bonus feat, the feat Improved Multiattack (Dungeon #109 49).
Just to be clear, according to Level Advancement—that may be optional at level 1 and that is kind of overly prescriptive anyway—when a creature advances a level, feats are picked before a creature has access to its spells and class features (PH 59).
In other words, to this reader, this technical ruling sounds unlikely.
Under laboratory conditions, technically Yes at level 3
Alternatively, it technically requires no DM intervention for a shapeshift druid upon advancing to level 3 to take the feat Multiattack, but doing so does require precise timing. While the druid is both in predator form and the subject of the spell beast claws, the druid must must gain enough XP to advance from level 2 to level 3. If those conditions are met then at that moment of advancement the druid meets the Multiattack feat's prerequisite and can take the feat. (This writer suggests seeking the DM's approval for this technique anyway as a DM may dislike it.)
In both cases—and others besides!—, when the beast claws spell ends or upon the druid exiting predator form, using the Multiattack feat would typically be impossible. Unless afterward the druid still had 3 or more natural weapons, the druid no longer meets the Multiattack feat's prerequisites. (If it's any consolation, the feat would still remain on the druid's character sheet despite the character not being able to use it.)
However, if they're really what you're after, ask the DM to make the feats available at level 1
With all that in mind, this DM's found that there's usually no real balance concerns with allowing a PC to pick a feat that the PC can benefit from only when the PC is using certain special abilities… except insofar as the PC may find himself sometimes—or even frequently—underpowered in comparison to his compatriots.
That is, were a player in this DM's campaign committed to this vision for his PC, this DM would just allow the level 1 shapeshift druid to take the feats without all the technical rigamarole: The combination is likely to be unexceptional at low levels when many things already have a small number of hit points anyway and the druid can only rarely bring his feats to bear. (Remember, not meeting a feat's prerequisites means not being unable to realize a feat's benefit at all!) And, at higher levels, such an accommodation means the druid's player won't have to game the system so as to acquire the feats he has in mind for his PC.