It is hard to answer, unfortunately as OP description is a little vague, a picture of both "take out" and "homemade" versions would be helpful.
Are you missing a certain fragrance, taste, texture, appearance, or all/some of the above?
We'll start by going through the ingredients you listed, or possibly missed out.
- Day old Rice
What kind of rice is used, and how was it cooked. Yes, this may sound crazy to some of you. However the difference in long grain rices will vary the taste and texture, not to mention moisture content of the dish.
I would suggest either a standard long grain rice of a jasmine rice (thai 'fragrant' jasmine rice is popular).
We will leave out the washing, and cooling of the cooked rice.
Two common ways of cooking is the 'unlimited water' (open boiling, like pasta) or 'limited water' (absorption).
I suggest the absorption method to maximise the flavour and reduce the moisture content.
It's not clear whether this is cooked or raw chicken or which cut it was.
I would suggest using a roast chicken and shredding the thigh (and the drumstick if required). This will give you a stronger chicken flavour with a better texture.
- Green Onions
Presuming this is cut into thin discs or thin slivers on the bias, and added to the rice seconds before serving.
Not my usual choice of ingredient in fried rice, but assuming these are:
Not much to say on eggs, presumably well beaten and stir fried/scrambled in a hot wok first.
- Soy Sauce
No clue if the fried rice is coloured and if so, how much. Light and/or Dark soy sauce may have been used.
As you say this is optional, there are other ways to add flavour (glutamates/umami) swiss bouillon powder for example, but this is unlikely to be used.
- Other Ingredients?
Suprisingly salt and oil have been omitted.
A fine salt would be the main seasoning ingredient in the dish apart from the chicken and spring onion.
A low tasting cooking oil with high smoking point would be best used. An optional light drizzle of a pure seasame oil (white sesame is cheaper compared to the stronger tasting black sesame).
- Cooking Method
Hypothetically based on a common industry standard mild steel wok using natural gas.
The wok is heated until smoking and cooking oil is added enough to scramble the eggs and prevent it from sticking. Chicken is added and heated before the rice is added. The rice will be left to fry until the heat sears one side slightly and then tossed before it sticks. Any clumps of Rice can be loosened before salt is added. Repeat searing and tossing, a few drops of water can be sprinkled if your rice is getting too dry and is not getting much heat transfer from the wok. Add your beansprouts, mix and heat them as you go.
Take a taste and check for temperature (75 degrees celsius if your in a customer serving environment) and seasoning (if you plan on adding light soy sauce, take this into consideration). Once you are happy, you can add your dark soy sauce in stages as you toss your rice to the colour you desire. Add your green onions for a minute or so. The heat will release their flavour as they start to wilt. You can also add sesame oil before your final tossing and serving.
Hopefully this brings you closer to the secret of your take out mystery.