To have all checkboxes checked by default, Navigate to Publishing section of About Us item and check all targets in Publishing targets field as shown below and then if you open the dialog Publishing Settings dialog, it would render all checkboxes in checked state.

Explanation -
Targets
tab in the Publishing Settings
screen is rendered by file \sitecore\shell\Applications\Content Manager\Dialogs\Set Publishing\Set Publishing.xml
It uses Sitecore.Shell.Applications.ContentManager.Dialogs.SetPublishing.SetPublishingForm
class from Sitecore.Client
dll
and here below is the function to render that tab
/// <summary>Renders the target tab.</summary>
/// <param name="item">The item.</param>
private void RenderTargetTab(Item item)
{
Assert.ArgumentNotNull((object) item, nameof (item));
Sitecore.Data.Fields.Field field = item.Fields[FieldIDs.PublishingTargets];
if (field == null)
return;
Item obj = Context.ContentDatabase.Items["/sitecore/system/publishing targets"];
if (obj == null)
return;
StringBuilder stringBuilder = new StringBuilder();
string str1 = field.Value;
foreach (Item child in obj.Children)
{
string str2 = str1.IndexOf(child.ID.ToString(), StringComparison.InvariantCulture) >= 0 ? " checked=\"true\"" : string.Empty;
string str3 = string.Empty;
if (this.ReadOnly)
str3 = " disabled=\"true\"";
stringBuilder.Append("<input id=\"pb_" + ShortID.Encode(child.ID) + "\" name=\"pb_" + ShortID.Encode(child.ID) + "\" class=\"scRibbonCheckbox\" type=\"checkbox\"" + str2 + str3 + " style=\"vertical-align:middle\"/>");
stringBuilder.Append(child.GetUIDisplayName());
stringBuilder.Append("<br/>");
}
this.PublishingTargets.InnerHtml = stringBuilder.ToString();
}
This function loops through all publishing targets and compares each target value on the current item's Publishing target field values to render the checkbox in a checked state.
Hope this answers your question.