We study integral points on the quadratic twists
E
D
:
y
2
=
x
3
−
D
2
x
\mathcal {E}_D:y^2=x^3-D^2x
of the congruent number curve. We give upper bounds on the number of integral points in each coset of
2
E
D
(
Q
)
2\mathcal {E}_D(\mathbb {Q})
in
E
D
(
Q
)
\mathcal {E}_D(\mathbb {Q})
and show that their total is
≪
(
3.8
)
rank
E
D
(
Q
)
\ll (3.8)^{\operatorname {rank} \mathcal {E}_D(\mathbb {Q})}
. We further show that the average number of non-torsion integral points in this family is bounded above by
2
2
. As an application we also deduce from our upper bounds that the system of simultaneous Pell equations
a
X
2
−
b
Y
2
=
d
,
b
Y
2
−
c
Z
2
=
d
aX^2-bY^2=d,\ bY^2-cZ^2=d
for pairwise coprime positive integers
a
,
b
,
c
,
d
a,b,c,d
, has at most
≪
(
3.6
)
ω
(
a
b
c
d
)
\ll (3.6)^{\omega (abcd)}
integer solutions.