No, you can not by Hyper Sonic in Sonic Rush.
For Sonic to go Hyper, he needs the 7 Super Emeralds, which Sonic Rush does not have. The Super Emeralds were only in Sonic 3 & Knuckles, so Sonic can only go hyper in Sonic 3 & Knuckles.
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Hyper Sonic and Super Tails are only availbale in Sonic 3 and Knuckles if you gather all chaos emeralds the super emeralds.
hyper sonic
Hyper Sonic only appeared in Sonic the Hedgehog 3 and Knuckles. Only Super Sonic makes an appearance in Sonic X.
there is super sonic, hyper sonic, werehog, and u might include metal sonic.