catch another process unhandled exception

You can try something like that to avoid the debugger question to appear, you won't get the exception but only the exit code:

class Program
{
    static void Main(string[] args)
    {
        try
        {
            ProcessStartInfo info = 
                 new ProcessStartInfo("ErroneusApp.exe");
            info.ErrorDialog = false;
            info.RedirectStandardError = true;
            info.RedirectStandardOutput = true;
            info.CreateNoWindow = true;
            info.UseShellExecute = false;

            System.Diagnostics.Process p = 
                System.Diagnostics.Process.Start(info);
            p.EnableRaisingEvents = true;
            p.Exited += p_Exited;
        }
        catch (Exception ex)
        {
            Console.WriteLine(ex.Message);
        }
        Console.ReadLine();
    }


    static void p_Exited(object sender, EventArgs e)
    {
        Process p = sender as Process;
        if (p != null)
        {
            Console.WriteLine("Exited with code:{0} ", p.ExitCode);
        }
        else
            Console.WriteLine("exited");
    }

}

In this question they provided another workaround for that, but changing some registry values.

If you are calling to a .Net executable assembly you can load it and (at your own risk :D ) call to the Main method of the Program class into a try_catch statement:

Assembly assembly = Assembly.LoadFrom("ErroneusApp.exe");
Type[] types= assembly.GetTypes();
foreach (Type t in types)
{
 MethodInfo method = t.GetMethod("Main",
     BindingFlags.Static | BindingFlags.NonPublic);
 if (method != null)
 {
    try
    {
        method.Invoke(null, null);
    }
    catch (Exception ex)
    {
        Console.WriteLine(ex.Message);
    }
    break;
 }
}

But be aware of the security risks you are introducing doing that.

No. If the controlled app utilizes standardError and return codes you may be notified of the occurance of an error or exception but you cannot trap it in any way.

Instead of doing the whole sneaking around the assembly to try to find the Main method like in jmservera's answer, you can simply execute the assembly in a new domain. See this msdn article