  The body-centered-tetragonal antiferromagnet EuGa$_4$ was recently identified
as a Weyl nodal-line semimetal that exhibits the topological Hall effect below
its reported antiferromagnetic (AFM) ordering temperature $T_{\rm N}= 15$-16.5
K which we find to be $T_{\rm N} = 16.4(2)$ K. The Eu$^{+2}$ ions are located
at the corners and body center of the unit cell. EuGa$_4$ exhibits A-type AFM
order below $T_{\rm N}$, where the Eu$^{2+}$ spin-7/2 moments are
ferromagnetically aligned in the $ab$ plane with the Eu moments in adjacent Eu
planes along the $c$ axis aligned antiferromagnetically. Low-field
magnetization versus field $M(H_{ab})$ data at $T=2$ K with the field aligned
in the $ab$ plane are reported that exhibit anomalous positive curvature up to
a critical field $H_{c1}$ at which a second-order transition occurs with
$H_{c1}\approx 0.85$ kOe for ${\bf H}\parallel [1,1,0]$ and $\approx 4.8$ kOe
for ${\bf H}\parallel [1,0,0]$. For larger fields, a linear behavior $M_{ab} =
\chi(T_{\rm N})H_{ab}$ is followed until the previously-reported critical field
$H^{\rm c}_{ab} = 71$ kOe is reached at which all moments become aligned with
the applied field. A theory is formulated for $T=0$ K that fits the observed
$M(H_{ab})$ behavior at $T=2$ K well, where domains of A-type AFM order with
fourfold rotational symmetry occur in the AFM state in zero field. The moments
in the four domains reorient to become almost perpendicular to ${\bf H}_{ab}$
at $H_{c1}$, followed by increasing canting of all moments toward the field
with increasing field up to $H^{\rm c}_{ab}$. A first-order transition in
$M(H_{ab})$ at $H_{ab}=H_{\rm c1}$ is predicted by the theory for $T=0$ K when
${\bf H}_{ab}$ is at a small angle from the [1,0,0] or [1,1,0] directions.
