Yes, but it is a minor development task, requiring a small, supported, edit to the Group - Banner widget.
The Group - Banner widget uses the $core_v2_ui.GetResizedImageHtml() API for server-side resizing of the group avatar. The method accepts an optional parameter, ResizeMethod, which supports two modes:
By default, the Group - Banner widget uses ZoomAndCrop, similar to user avatars. The Group - Banner also uses CSS to make the resized image round.
To edit the Widget:
Administration > Interface > Widgets > Group - Banner > Content
Replace the following block:
<div class="avatar">
<a href="$core_v2_encoding.HtmlAttributeEncode($group.Url)" #if ($isRegistered) $core_v2_tourTip.Render($core_v2_language.GetResource('TourTipTitle')) #end >
$core_v2_ui.GetResizedImageHtml($core_v2_group.Current.AvatarUrl, 68, 68, "%{ alt = $group.Name, ResizeMethod='ZoomAndCrop' }")
</a>
</div>
with
<div class="avatar" style="border-radius:0">
<a href="$core_v2_encoding.HtmlAttributeEncode($group.Url)" #if ($isRegistered) $core_v2_tourTip.Render($core_v2_language.GetResource('TourTipTitle')) #end >
$core_v2_ui.GetResizedImageHtml($core_v2_group.Current.AvatarUrl, 68, 68, "%{ alt = $group.Name, ResizeMethod='ScaleDown', style='border-radius:0;' }")
</a>
</div>
It's worth noting that changing this widget changes it for all groups. So you may find it preferable to first save the widget as a separate copy, and make the change to that widget instead. And then the copied widget would need to be applied to the Group's header in page management.
An alternative would be to make you image square by padding it with whitespace. (e.g. if your image is 100px high and 200px wide, add 50px of white to the top & the bottom to make it 200x200px)