So, what concerns do you have?
Two simple equestions and two quotas.

Example.
1. equation (SO1) (hidden) after question Q1
{if(Q1_1!="Y", 0,1)}

2. equation (SO2) (hidden) after question Q2
{if (Q2 <= 24 OR Q2 >= 55,0,1)}

3. First quota
 
 

4. Second quota analog.

That's all
Create your end pages however you want.
 
 

 

I really don't see how this requires a submit button, if they don't make it through the quota, their survey is "submitted" automatically.

I don't see how two different buttons can help in this situation at all.

If these are two seperate quotas, create two zero quotas based on the criteria you need. If they are interlocked, you can create an equation question that checks if they should be screened out based on the two criteria and then set the quota on the two questions.

If you really, really want an additional submit button (which would mean that your respondents that are actually supposed to be screened out are counting into the 'completed' sample, which I don't think makes sense), then I am out. Because I don't know how to have an additional submit button in the middle of the survey.

No, his idea was to display the screenout message in a  following text display and then make the button jump to the end.

In my opinion the TO isn't familiar at all with quotas.
 

In my opinion the TO isn't familiar at all with quotas.

That is also my impression. Because for each quota, there is a specific end page where you can leave specific messages. No need to create a text display question. You can write your text as Quota message.

Because the way he wants to go about (if it is the case what you are saying, Joffm), then there are a couple of downsides to his approach:

1. If you want to "submit" a survey that is actually a screen out, it is counted as a completed survey in Limesurvey.
2. You would need to create conditions for all subsequent questions to hide them

@OP: I really think you should make yourself familiar with quotas and especially zero quotas, what they do, how they work. I think you will find that they are what you are looking for all along.