Can every DFA be converted to an equivalent one that has a single accept state
Myhill&ndash;Nerode theory implies the following result:


  Every DFA for a regular language has at least as many accepting states as the minimal DFA.


Indeed, suppose that the minimal DFA has $m$ accepting states. Let $w_1,\ldots,w_m$ be words which cause the minimal DFA to reach these states. Now consider any DFA for the same language, and let $q_1,\ldots,q_m$ be the states that the DFA reaches upon reading $w_1,\ldots,w_m$. Since $w_1,\ldots,w_m$ are in the language, all these states must be accepting. Furthermore, they must be different, since they are pairwise inequivalent (for each $i \neq j$ there exists a word $x$ such that $w_ix$ is in the language and $w_jx$ isn't, or vice versa). So the DFA contains at least $m$ accepting states.

It is not difficult to construct a regular language whose minimal DFA contains more than one accepting state - for example, the language of all words whose length is not divisible by 3. This language refutes your conjecture.