  Linear optical realizations of Bell state measurement (BSM) on two
single-photon qubits succeed with probability $p_s$ no higher than $0.5$.
However pre-detection quadrature squeezing, i.e., quantum noise limited phase
sensitive amplification, in the usual linear-optical BSM circuit, can yield
${p_s \approx 0.643}$. The ability to achieve $p_s > 0.5$ has been found to be
critical in resource-efficient realizations of linear optical quantum computing
and all-photonic quantum repeaters. Yet, the aforesaid value of $p_s > 0.5$ is
not known to be the maximum achievable using squeezing, thereby leaving it open
whether close-to-$100\%$ efficient BSM might be achievable using squeezing as a
resource. In this paper, we report new insights on why squeezing-enhanced BSM
achieves $p_s > 0.5$. Using this, we show that the previously-reported ${p_s
\approx 0.643}$ at single-mode squeezing strength $r=0.6585$---for unambiguous
state discrimination (USD) of all four Bell states---is an experimentally
unachievable point result, which drops to $p_s \approx 0.59$ with the slightest
change in $r$. We however show that squeezing-induced boosting of $p_s$ with
USD operation is still possible over a continuous range of $r$, with an
experimentally achievable maximum occurring at $r=0.5774$, achieving ${p_s
\approx 0.596}$. Finally, deviating from USD operation, we explore a
trade-space between $p_s$, the probability with which the BSM circuit declares
a "success", versus the probability of error $p_e$, the probability of an input
Bell state being erroneously identified given the circuit declares a success.
Since quantum error correction could correct for some $p_e > 0$, this tradeoff
may enable better quantum repeater designs by potentially increasing the
entanglement generation rates with $p_s$ exceeding what is possible with
traditionally-studied USD operation of BSMs.
