I forgot towel day

Python 2, 223 210 209 204 bytes

def f(s,m):w=12*s-2;l='    |';print'[===|'+' '*w+'|====]';exec"j=(m+' ').rfind(' ',0,-~w);print l+'%*s||'%(w,m[:j]);m=m[j+1:];"*s*5;print(l+w*'='+'||\n')*2+' '*4+'"'*12*s+'|\n '+l+' '*w+'|\n    ','"'*12*s

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JavaScript (Node.js), 375 bytes

This is the worst submission you will get, but at least I tried xD half of bytes are spaces though

(s,_,w=12*s-2,h=5*s)=>`
[===|${j=' '.repeat(w)}|====]
${[...Array(h)].map((g,i)=>_.split` `.reduce((a,b)=>((l=a.split`,`)[l.length-1]+b).length>w-1?a+','+b:a+' '+b,'').split`,`[i]).map(a=>`    |${a?(r=a.length)<w?a+' '.repeat(w-r):a:j}||    `).join`\n`+
`
    |${'='.repeat(w)}||    `.repeat(2)}
    ${y='"'.repeat(w+2)}|    
     |${j}|    
     ${y}
`

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JavaScript (Node.js), 347 345 343 337 334 328 326 bytes

b=>d=>`
[===|${M=" "[X="repeat"](c=12*b-2)}|====]
${[...Array(5*b)].map((a,b)=>d[Y="split"]` `.reduce((a,b)=>((l=a[Y]`,`)[l[Z="length"]-1]+b)[Z]>c-1?a+","+b:a+" "+b)[Y]`,`[b]).map(a=>`    |${a?(r=a[Z])<c?a+" "[X](c-r):a:" "[X](c)}||    `).join`
`+`
    |${"="[X](c)}||    `[X](2)}
    ${y='"'[X](c+2)}|
     |${M}|
     ${y}
`

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---

Explanation :

b =>                                   // lambda function taking arg 1 : size
    d =>                               // arg 2 : message
        `                              // begin string template for drawing
[===|${                                // draw first part and now open literal 
        M = ' '[X='repeat'(c=12*b-2)]  // set 3 variables, M, X , c to be used again
    }|====]                            // close and draw the next part ====]
${[...Array(5*b)]                      // open and create an array of length 5 * b =width
.map(a,b=>                             // begin map function two args : a,b
        d[Y='split']` `                // use d(message), split at whitespace, set to Y
.reduce((a,b) =>                       // reduce to single value, arg 1 : a, arg 2 : b
        ((l = a[Y]`,`)                 // declare l and then find in l
        [l[Z='length']-1]              // set Z as length
        + b)                           // add value of b
        [Z]                            // find the length 
        > c-1 ?                        // check if it's less than c - 1
        a+','+b                        // then add `${a},${b}`
        : a + ' ' + b                  // otherwise `${a} ${b}`
        )[Y]`,`                        // close and split at comma
        [b]                            // use b again
        )                              // close
        .map(a =>                      // map over that arg 1 : a
                `    |${               // four space + | and open 
        a ?                            // if a is true or a truthy value
            (r=a[Z])                   // set value of r as a's length
            < c ?                      // check if it's less than c
                a+' '[X](c-r)          // then draw a + space repeated c-r times
                : a + ' '[X](c)        // else draw a + space repeated c times
            }                          // close
            ||    `                    // add || and 4 spaces and close
        )                              // end
        .join`
`                                      // and turn to string with new line as separator 
    +                                  // add to that
    `
    |{                                 // new line , 4 spaces and | 
        '='[X](c)}                     // repeat = c times
        ||    `[X](2)}                 // and repeat that 2 times
    ${                                 // new line + 4 space
    y = '"'[X](c+2)                    // repeat " c + 2 times and set to y
    }|                                 // close and add |
    |{                                 // add | and open
        M}|                            // put M and close and add |
    {y}                                // new line , 4 spaces and variable y
`                                      // end with new line.