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Euclidean $n$-component $\phi^4$ theories whose Hamiltonians are O(n)symmetric except for quadratic symmetry breaking boundary terms are studied infilms of thickness $L$. The boundary terms imply the Robin boundary conditions$\partial_n\phi_\alpha =\mathring{c}^{(j)}_\alpha \phi_\alpha $ at the boundaryplanes $\mathfrak{B}_{j=1,2}$ at $z=0$ and $z=L$. Particular attention is paidto the cases in which $m_j$ of the $n$ variables $\mathring{c}^{(j)}_\alpha$take the special value $\mathring{c}_{m_j\text{-sp}}$ corresponding to criticalenhancement while the remaining ones are subcritically enhanced. Under theseconditions, the semi-infinite system bounded by $\mathfrak{B}_j$ has amulticritical point, called $m_j$-special, at which an $O(m_j)$ symmetriccritical surface phase coexists with the O(n) symmetric bulk phase, provided$d$ is sufficiently large. The $L$-dependent part of the reduced free energyper area behaves as $\Delta_C/L^{d-1}$ as $L\to\infty$ at the bulk criticalpoint. The Casimir amplitudes $\Delta_C$ are determined for small$\epsilon=4-d$ in the general case where $m_{c,c}$ components $\phi_\alpha$ arecritically enhanced at both boundary planes, $m_{c,D} + m_{D,c}$ components areenhanced at one plane but satisfy asymptotic Dirichlet boundary conditions atthe respective other, and the remaining $m_{D,D}$ components satisfy asymptoticDirichlet boundary conditions at both $\mathfrak{B}_j$. Whenever $m_{c,c}>0$,these expansions involve integer and fractional powers $\epsilon^{k/2}$ with$k\ge 3$ (mod logarithms). Results to $O(\epsilon^{3/2})$ for general values of$m_{c,c}$, $m_{c,D}+m_{D,c}$, and $m_{D,D}$ are used to estimate the $\Delta_C$of 3D Heisenberg systems with surface spin anisotropies when $(m_{c,c},m_{c,D}+ m_{D,c}) = (1,0)$, $(0,1)$, and $(1,1)$.

Critical Casimir amplitudes for $n$-component $\phi^4$ models with O(n)-symmetry breaking quadratic boundary terms