The statement, "Stone walls do not a prison make, nor iron bars a cage" implies that physical imprisonment is not the only form of prison that one can suffer. There are two meanings to this idea, the first being that a physical prison will not necessarily hold you—the stone walls will not be a prison. In the case of Tuck and his family, this is somewhat true—as they have been executed and imprisoned but, due to their immortality, they survive and continue to live on.

The reverse point of this statement is that they are in an altogether different sort of prison—a bondage to this physical earth. There are chains that are metaphysical in nature that are adhering them to life on Earth when, many times, they would prefer to perish. Not all cages are made of stone or iron, and for Tuck's family, theirs is made of their immortality.

Those lines are the first two lines from stanza four of Richard Lovelace's poem "To Althea, from Prison."  In the poem, Lovelace is using those lines to explain that his imagination, soul, and love are free from any sort of physical prison made by men.  

Those two lines do appear in the short novel Tuck Everlasting, but Winnie's use of the quote is a bit different from Lovelace's poem.  In Chapter 24, Winnie sneaks out of her house in order to help the Tucks rescue Mae from jail.  When Winnie sees the jail and its barred windows, those two lines pop into her head. 

Here, too high for Winnie to see into, was a barred window through which, from the room in front, light glowed faintly. Winnie peered up at it, at the blackness of the bars with the dim gold of the light between. Into her head came lines from an old poem:

Stone walls do not a prison make,

Nor iron bars a cage.

Winnie doesn't know why the lines come to her, but the lines do make sense in Mae's situation.  Mae most definitely is in a prison with iron bars.  She is caged.  But the prison is only a physical cage for her.  What would really make a prison and cage for Mae is not a physical building.  What would utterly cage and destroy her soul and love is separation from her family.  If Mae can't be broken out of jail, her immortal secret will become known.  She will become a freak of nature that everybody else wants to poke, prod, and test.  She would never be able to see her family again . . . forever.  That would be a far greater prison to Mae than any actual four-walled room with iron bars in a window.