As barbara beeton mentioned in a comment, flexysym and eulervm don't cooperate. A possible workaroung is to use the savesym package: the idea is to load breqn and then rename the conflicting symbols (the whole Greek alphabet and some others) using \savesymbol. Once this is done, eulervm can be safely loaded.
\documentclass{article}
\usepackage[T1]{fontenc}
\usepackage{breqn}
\usepackage{savesym}
\savesymbol{Gamma}
\savesymbol{Delta}
\savesymbol{Theta}
\savesymbol{Lambda}
\savesymbol{Xi}
\savesymbol{Pi}
\savesymbol{Sigma}
\savesymbol{Upsilon}
\savesymbol{Phi}
\savesymbol{Psi}
\savesymbol{Omega}
\savesymbol{alpha}
\savesymbol{beta}
\savesymbol{gamma}
\savesymbol{delta}
\savesymbol{epsilon}
\savesymbol{zeta}
\savesymbol{eta}
\savesymbol{theta}
\savesymbol{iota}
\savesymbol{kappa}
\savesymbol{lambda}
\savesymbol{mu}
\savesymbol{nu}
\savesymbol{xi}
\savesymbol{pi}
\savesymbol{rho}
\savesymbol{sigma}
\savesymbol{tau}
\savesymbol{upsilon}
\savesymbol{phi}
\savesymbol{chi}
\savesymbol{psi}
\savesymbol{omega}
\savesymbol{varepsilon}
\savesymbol{vartheta}
\savesymbol{varpi}
\savesymbol{varphi}
\savesymbol{infty}
\savesymbol{ldotp}
\usepackage{eulervm}
\begin{document}
\[ \Sigma_1\Delta_2 \]
\end{document}

A more definitive solution would be to create a support file in flexysim for eulervm; this doen't seem to be so hard according to the documentation of breqn:
If you find that a particular math
font package doesn’t work then please
see implementation in flexisym.dtx
for how to create a support file—it is
easier than one might think.
Contributions welcome.
eulerandeulervmhave a working$\Sigma$for me... – Seamus Jun 16 '11 at 14:22{}button. – Jake Jun 16 '11 at 14:43\Sigmabut the whole greek alphabet, making the obvious adjustment of reversing the order of the\usepackagecommands inoperative. i think this is not a bug, but a design decision, and would have to ask the maintainer of breqn about it, if it's not obvious from the.dtxfile. it's not possible to ask the original author as he died in 2003. however, redefinitions of the greek letters were in a 2002 version of breqn, although in a separate file, not in `breqn.sty' itself. – barbara beeton Jun 16 '11 at 15:36