Is it possible for a U.S. citizen to evade income tax by being paid in Bitcoin?
No. You have to pay taxes no matter what currency you are paid in, or for that matter, in most barter transaction as well. You would have to pay taxes even if you were paid in goats.
Also, the counter-parties in transactions in which you are paid in Bitcoin and earn money often have an obligation to file information tax returns to the IRS.
If you don't report the income, both you and any counter-party subject to reporting requirements could be liable for the tax as well as for penalties and interest and possible criminal charges as well for intentionally evading taxation and filing false tax returns.
Taxable income (Merriam-Webster) as defined by the IRS is
...generally speaking, is the gross income of an individual or corporation, less any allowable tax deductions. Your taxable income is, in other words, the amount of your income that is subject to income tax. What is taxable income? | US Tax Center
So if you have a good accountant, you may not have to pay much or all on your income depending on your deductions. But that's more of an accounting question.
But income that is paid in Bitcoin is still income; it doesn't matter if you keep the funds in Bitcoin or convert them to US dollars, it's still income. Even in Idaho.
And the people and/or businesses who pay you have to keep tax records and report your income. For annual amounts less than $600, Form 1099-MISC doesn't need to be filed by the employer, but the payee still needs to show the income on their tax return.
The web has many accounts of people who fall into the trap of thinking that paying income taxes is illegal because tax policies are not clearly outlined in the Constitution. And they may also feel that Bitcoin itself is a currency outside the realm of the law, and when that earned Bitcoin is not converted to a traditional currency, such as the US dollar, it is outside the law. But if that Bitcoin is income as defined in the dictionary, it is subject to income taxes. People who evade paying income taxes run the risk of an audit and prosecution for not paying taxes. Read Tax protester statutory arguments - Wikipedia.