If I am unable t pay off my education loans because I cannot work due to illness, can the government try to get the money from my wife's income?

No, they cannot directly garnish her wages assuming she is not a co-signer of your loan. However, one technique they use it to take some or all of any income tax refund or other federal benefits such as social security, so if you file your income taxes jointly they could get at what is essentially her refund that way. Once it goes to a collection agency if you are sued you can expect possible seizing of property or assets such as your checking account, so if you have a joint account then her wages could be gotten at that way.

Are you sure about Social Security? My understanding was they cold not touch Social Security.

The government is authorized by congress to be very aggressive with respect to recovering delinquent student loans, and is able to take from any government benefits - including social security. It's different with defaults on private loans (mortgages and credt cards and such) where your SS benefits are shielded.