No they don't.
Here's one without an x-intercept.
The y-intercept is where x= 0. Since evaluating a polynomial involves only multiplication and addition or subtraction, which are possible for any real number, the domain of a polynomial always includes all real numbers, including 0.
The x-intercept is where y, the value of the polynomial, is 0. Any polynomial of odd degree crosses the x-axis and so has an x-intercept. Polynomials of even degree may or may not have an x-intercept.