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I'm pretty sure this is an open question--or at least I've heard it referred to as such during this calendar year by multiple people who work on C^0 symplectic topology.
I think it follows pretty quickly from the jet transversality theorem (see e.g. Hirsch's differential topology book) that if $n>3$ generic embeddings $f:M\to R^{2n}$ have no points at which they are Lagrangian. Indeed the Lagrangian Grassmannian has, by my count, codimension $n(n-1)/2$ in the whole Grassmannian $Gr(2n,n)$, and so if $n(n-1)/2>n$ then the "expected dimension" of the Lagrangian locus is negative, and jet transversality implies that generic perturbations of a given map will realize this expected dimension. So that leaves only the case $n=3$ unresolved.
No, that certainly wouldn't be the case (for instance if you composed the symplectomorphism in the answer with a non-equivariant symplectomorphism of the domain--of which there are many--the result wouldn't be equivariant). The idea of asking for O(n)-equivariance was just to narrow down the search, and also to take advantage of the fact that an equivariant symplectomorphism will respect the moment maps (thus giving a function that the map should preserve, which is usually easier to arrange than preserving a form).
