My question is rather clear: Is every spontaneous symmetry breaking connected with a second order phase transition?
In many books (A.Zee QFT in a nutshell or also in P&S etc.) when they start explaining spontaneous symmetry breaking, the case of loss of ferromagnetism at the Curie temperature is mentioned as an example. This transition from a state with non-zero directed magnetisation to zero magnetisation is a well-known example of second-order phase transition. Bose-Einstein condensation is also considered as a second order phase transition. A.Zee describes in his book QFT in a nutshell that superfluidity is a process of spontaneous symmetry breaking with a non-relativistic boson-field with Mexican-hat potential and explains even the emergence of Nambu-Goldstone bosons in case of the symmetry breaking. I assume that in case of a gauge field symmetry breaking the situation might be a bit different.
It is also rather remarkable that according to the Landau-theory of second-order transitions the dependence of Gibbs potential on the magnetisation looks very much like a Mexican potential in one dimension (see fig. 8.3 of P&S) for the case $T<T_c$.