Your bonus is taxed at exactly the same rate as the rest of your salary, its just the withholding that differs (because the bonus isn't fitted into the regular withholding calculations). It will all be normalized on your annual 1040.
Is there a tax advantage to putting 100% of my bonus into 401k?
No. You can adjust your withholding rate using updated form W4, to account for the extra withheld because of the bonus.
From looking at my pay stubs, my bonus gets taxed at a higher rate
than my regular paychecks.
Bonus tax withholding can be upwards of 40% after federal (25%) and state taxes. For some its higher rate, for others it may in fact be lower.
Federal withholding for lump-sum payments is usually flat 25% (usually, because the employers can, if they want to, adjust their withholding calculations to account for the bonus, and then withhold the "regular" amount. But it usually complicates the accounting and most chose not to). Similarly, State withholding is probably calculated at some State-mandated high (closer to the highest bracket) rate. That makes the lump-sum tax calculation simple, but for most people - the withholding will be higher than the actual tax liability. For some (high earners) it will be the opposite - the flat-rate withholding will be lower than their marginal tax and they'll end up underpaying their taxes.
And since traditional 401k contributions are pre-tax, the thinking is
that contributions from my bonus paycheck will save me more in taxes
than contributions from my regular paychecks.
No, because the 25% is the withholding, not the actual tax. The actual tax on your whole compensation is calculated using your marginal rates. You'll get the actual amount when you fill your tax return. If because of the extra withholding on your bonus you ended up paying more than you should have - you'll get a refund from the IRS.
Here's an article on askcpa on this exact question.