We show that Zorboska’s criterion for compactness of Toeplitz operators with
BMO
1
\text {BMO}^1
symbols on the Bergman space of the unit disc holds, by a different proof, for the Segal-Bargmann space of Gaussian square-integrable entire functions on
C
n
\mathbb {C}^n
. We establish some basic properties of
BMO
p
\text {BMO}^p
for
p
≥
1
p \geq 1
and complete the characterization of bounded and compact Toeplitz operators with
BMO
1
\text {BMO}^1
symbols. Via the Bargmann isometry and results of Lo and Englis̆, we also give a compactness criterion for the Gabor-Daubechies “windowed Fourier localization operators” on
L
2
(
R
n
,
d
v
)
L^2(\mathbb {R}^n, dv)
when the symbol is in a
BMO
1
\text {BMO}^1
Sobolev-type space. Finally, we discuss examples of the compactness criterion and counterexamples to the unrestricted application of this criterion for the compactness of Toeplitz operators.