Consider two boson of spin $1$ without angular momentum. 
I'm seeing an argument that "because those two particles were boson, they must be symmetric under the exchange $m_1,m_2$. Thus they could not be in $J=1$ states such as $\frac{1}{\sqrt{2}}(|1,0\rangle+|0,1\rangle$)". 

From appearance it kind of make sense, but then it doesn't, i.e. what if the spacial part of the wave function was asymmetric as well? Doesn't that resolve the issue?

Why two spin $1$ boson could not be at spin $|1,0\rangle-|0,1\rangle$ states?