continue shopping button forwards to wrong page

The option "Continue shopping button's URL" if only for the continue button on the checkout.Not for the one in the popup. The one in the popup should just close the popup.
What's happening is that you have a javascript error on the page and when you click on the link, it can't close the popup and just display the notice view.

Yes. Edit the file notice of the view checkout via the menu Display->Views and change the line:
echo $this->cartClass->displayButton(JText::_('CONTINUE_SHOPPING'),'continue_shopping',$this->params,'','try{ window.top.document.getElementById(\'sbox-window\').close(); }catch(err){ window.top.SqueezeBox.close(); } return false;','id="hikashop_add_to_cart_continue_button"');

to:
echo $this->cartClass->displayButton(JText::_('CONTINUE_SHOPPING'),'continue_shopping',$this->params,'','window.top.location=\' mywebsite.com/mylink\';return false;','id="hikashop_add_to_cart_continue_button"');

Try like that:
echo $this->cartClass->displayButton(JText::_('CONTINUE_SHOPPING'),'continue_shopping',$this->params,'','window.top.location=\'mywebsite.com/mylink\';return false;','id="hikashop_add_to_cart_continue_button"');

Of course, you need to replace the text mywebsite.com/mylink by your link...

You need to add http:// at the beginning of your link, otherwise the SEF system of joomla prepends the url of your page to your link.

Well, you can add https:// at the beginning then.

Then you should just change the code to use the current protocol...like that:
echo $this->cartClass->displayButton(JText::_('CONTINUE_SHOPPING'),'continue_shopping',$this->params,'','window.top.location=\'http'.(hikashop_isSSL()? 's' : '').'://mywebsite.com/mylink\';return false;','id="hikashop_add_to_cart_continue_button"');

Maybe.
You can try that:
echo $this->cartClass->displayButton(JText::_('CONTINUE_SHOPPING'),'continue_shopping',$this->params,'','window.top.location=\''.JRoute::_('index.php?option=com_content').'\';return false;','id="hikashop_add_to_cart_continue_button"');
where index.php?option=com_content is your URL without the website URL.
But if that doesn't work, then it's not possible.

The continue shopping button on the checkout uses the URL that you set in the option "Continue shopping button's URL" of the configuration.
You can set it to index.php for example and it will work on any domain name.