Spontaneous symmetry breaking (SSB or SBS) refers to the fact that the lowest energy state, the vacuum, may not be invariant under all symmetries of the
theory, in other words, several vacua are possible. While it was always a logical possibility, before 1950s physicists generally regarded it as unphysical. First examples of SBS did indeed appear in condensed
matter physics, and the idea was transplanted into particle physics by analogy (as Nambu put it, "I must say that my early exposure to condensed matter physics has been quite beneficial to me"). Jona-Lasinio gives some historical details in Spontaneous symmetry breaking: some history and some variations on the theme:
"Heisenberg, Z. Naturforsch.14, 441 (1959), Proceedings of the 1960 Rochester Conference p. 851, was the first to consider SBS as a possibly relevant concept in particle physics... The theory of superconductivity of Bardeen, Cooper and Schrieffer which appeared in 1957 provided the key paradigm for the introduction of SBS in relativistic quantum field theory on the basis of an analogy proposed by Nambu, Phys. Rev. Lett. 4, 380 (1960)... The BCS theory of superconductivity was reformulated and developed by various authors including Bogolubov, Valatin, Anderson, Ricayzen and Nambu. The paper of Nambu, Phys. Rev.
117, 648 (1960), used a language akin to quantum field theory, that is Green’s functions formalism."
As for "right" and "wrong" problems, it is always easy to make such judgements with a benefit of hindsight. Yang and Mills only introduced their non-Abelian gauge theories in 1956, much about them was still obscure and explored in the early 60s. The generality of the Goldstone "theorem", and of the massless Nambu-Goldstone bosons it predicted, was by no means clear, and neither was the "right" way to avoid them. Giving up the gauge principle altogether was considered as an option, along with questioning "theorem's" generality. In fact, it is these "wrong" bosons that led to the invention of the "right" Higgs mechanism. Karaca's Construction of the Higgs Mechanism and the Emergence of the Electroweak Theory gives a detailed account:
"This led, among the quantum field theorists, to the supposition that massless scalar bosons would be an inevitable consequence of all dynamical models of the Yang-Mills theory; thereby presenting a supposed dilemma which can be expressed as follows . The only way to solve the zero - mass problem and give mass to gauge bosons is through the SSB of the original gauge symmetry of the Lagrangian. However, in this way, one encounters the difficulty posed by the Goldstone theorem — often referred to as the “Goldstone zero - mass difficulty” — namely that, the existence of massless scalar bosons for which there is no experimental evidence...
The first explicit objection against the general validity of the Goldstone theorem came in 1963 from a solid-state physicist, namely, Philip Anderson... However, Anderson’s argument was an analogy argument from solid-state physics, and the demonstration of its validity within the theoretical framework of the Yang-Mills theory awaited contributions by a number of physicists... Higgs showed that the Goldstone theorem would break down in a class of non-manifestly Lorentz-covariant gauge theories that use the “radiation gauge”...