Get icon from website, show and update icon
I'd restructure this code significantly. The first thing I'd do is move the image-retrieval into a separate method and the image placing would then be "simpler"..

Compare your code against:

Image image = getRemoteLogo();
if (image != null) {
    // Logo updated
    updateLogo(image);
}


This is largely reimagining what you're doing, and depending on how you prefer it, you might move the nullcheck into updateLogo.

This allows you to hide the image and get rid of the IMO .. clunky null-initialization before the try-block:

private Image getRemoteLogo() {
    try {
       URL remoteUrl = new URL(Config.getRemoteLogoUrl());
       return Toolkit.getDefaultToolkit().getImage(remoteUrl);
    } catch (MalformedURLException ex) {
       // this is the point where you can slap the programmer :D
       ex.printStackTrace();
       return null;
    }
}


This neatly hides the hardcoded dropbox URL in a (not further specified) Configuration, which allows changing the logo location without having to replace classes.

Also it cleanly abstracts away the Image-loading and has clear behavior in error cases. It also gets rid of the superfluous boolean flag you have.

The updateLogo(Image) is history then, you do your ComputerCheck and set the image accordingly.

The startup slowdown is something you'll have to live with.

Oh and I changed the variable names slightly. Use speaking names wherever possible, future maintainers (including yourself) may not be able to map out which number corresponds to which URL :)